Intuitive Guide to Angles, Degrees and Radians

Wrong. Most of us have no idea why there’s 360 degrees in a circle. We memorize a magic number as the “size of a circle” and set ourselves up for confusion when studying advanced math or physics, with their so called “radians”.

“Radians make math easier!” the experts say, without a simple reason why (discussions involving Taylor series are not simple). Today we’ll uncover what radians really are, and the intuitive reason they make math easier.

Where Do Degrees Come From?

Before numbers and language we had the stars. Ancient civilizations used astronomy to mark the seasons, predict the future, and appease the gods (when making human sacrifices, they’d better be on time).

How is this relevant to angles? Well, bub, riddle me this: isn’t it strange that a circle has 360 degrees and a year has 365 days? And isn’t it weird that constellations just happen to circle the sky during the course of a year?

Unlike a pirate, I bet you landlubbers can’t determine the seasons by the night sky. Here’s the Big Dipper (Great Bear) as seen from New York City in 2008 (try any city):

Constellations make a circle every day (video). If you look at the same time every day (midnight), they will also make a circle throughout the year. Here’s a theory about how degrees came to pass:

Humans noticed that constellations moved in a full circle every year

Every day, they were off by a tiny bit (“a degree”)

Since a year has about 360 days, a circle had 360 degrees

But, but… why not 365 degrees in a circle?

Cut ‘em some slack: they had sundials and didn’t know a year should have a convenient 365.242199 degrees like you do.

360 is close enough for government work. It fits nicely into the Babylonian base-60 number system, and divides well (by 2, 3, 4, 6, 10, 12, 15, 30, 45, 90… you get the idea).

Basing Mathematics on the Sun Seems Perfectly Reasonable

Earth lucked out: ~360 is a great number of days to have in a year. But it does seem arbitrary: on Mars we’d have roughly ~680 degrees in a circle, for the longer Martian year. And in parts of Europe they’ve used gradians, where you divide a circle into 400 pieces.

Many explanations stop here saying, “Well, the degree is arbitrary but we need to pick some number.” Not here: we’ll see that the entire premise of the degree is backwards.

Radians Rule, Degrees Drool

A degree is the amount I, an observer, need to tilt my head to see you, the mover. It’s a tad self-centered, don’t you think?

Suppose you saw a friend go running on a large track:

“Hey Bill, how far did you go?”

“Well, I had a really good pace, I think I went 6 or 7 mile–”

“Shuddup. How far did I turn my head to see you move?”

“What?”

“I’ll use small words for you. Me in middle of track. You ran around. How…much…did…I…turn…my…head?”

“Jerk.”

Selfish, right? That’s how we do math! We write equations in terms of “Hey, how far did I turn my head see that planet/pendulum/wheel move?”. I bet you’ve never bothered to think about the pendulum’s feelings, hopes and dreams.

Do you think the equations of physics should be made simple for the mover or observer?

Radians: The Unselfish Choice

Much of physics (and life!) involves leaving your reference frame and seeing things from another’s viewpoint. Instead of wondering how far we tilted our heads, consider how far the other person moved.

But absolute distance isn’t that useful, since going 10 miles is a different number of laps depending on the track. So we divide by radius to get a normalized angle:

You’ll often see this as

or angle in radians (theta) is arc length (s) divided by radius (r).

A circle has 360 degrees or 2pi radians — going all the way around is 2 * pi * r / r. So a radian is about 360 /(2 * pi) or 57.3 degrees.

Now don’t be like me, memorizing this thinking “Great, another unit. 57.3 degrees is so weird.” Because it is weird when you’re still thinking about you!

Moving 1 radian (unit) is a perfectly normal distance to travel. Put another way, our idea of a “clean, 90 degree angle” means the mover goes a very unclean pi/2 units. Think about it — “Hey Bill, can you run 90 degrees for me? What’s that? Oh, yeah, that’d be pi/2 miles from your point of view.” The strangeness goes both ways.

Radians are the empathetic way to do math — a shift from away from head tilting and towards the mover’s perspective.

What’s in a name?

Radians are a count of distance in terms of “radius units”, and I think of “radian” as shorthand for that concept.

Strictly speaking, radians are just a number like 1.5 or 73, and don’t have any units (in the calculation “radians = distance traveled / radius”, we see length is divided by length, so any units would cancel).

But practically speaking, we’re not math robots, and it helps to think of radians as “distance” traveled on a unit circle.

Using Radians

I’m still getting used to thinking in radians. But we encounter the concept of “mover’s distance” quite a bit:

We use “rotations per minute” not “degrees per second” when measuring certain rotational speeds. This is a shift towards the mover’s reference point (“How many laps has it gone?”) and away from an arbitrary degree measure.

When a satellite orbits the Earth, we understand its speed in “miles per hour”, not “degrees per hour”. Now divide by the distance to the satellite and you get the orbital speed in radians per hour.

Sine, that wonderful function, is defined in terms of radians as

This formula only works when x is in radians! Why? Well, sine is fundamentally related to distance moved, not head-tilting. But we’ll save that discussion for another day.

Radian Example 1: Wheels of the Bus

Let’s try a real example: you have a bus with wheels of radius 2 meters (it’s a monster truck bus). I’ll say how fast the wheels are turning and you say how fast the bus is moving. Ready?

Radians are distance along a unit circle — we just scale by the real radius to see how far we’ve gone. 6 * 2 = 12 meters per second. Next question.

Wow! No crazy formulas, no pi floating around — just multiply to convert rotational speed to linear speed. All because radians speak in terms of the mover.

The reverse is easy too. Suppose you’re cruising 90 feet per second on the highway (60 miles per hour) on your 24″ inch rims (radius 1 foot). How fast are the wheels turning?

Well, 90 feet per second / 1 foot radius = 90 radians per second.

That was easy. I suspect rappers sing about 24″ rims for this very reason.

Radian Example 2: sin(x)

Time for a beefier example. Calculus is about many things, and one is what happens when numbers get really big or really small.

Choose a number of degrees (x), and put sin(x) into your calculator:

When you make x small, like .01, sin(x) gets small as well. And the ratio of sin(x)/x seems to be about .017 — what does that mean? Even stranger, what does it mean to multiply or divide by a degree? Can you have square or cubic degrees?

Radians to the rescue! Knowing they refer to distance traveled (they’re not just a ratio!), we can interpret the equation this way:

x is how far you traveled along a circle

sin(x) is how high on the circle you are

So sin(x)/x is the ratio of how high you are to how far you’ve gone: the amount of energy that went in an “upward” direction. If you move vertically, that ratio is 100%. If you move horizontally, that ratio is 0%.

When something moves a tiny amount, such as 0 to 1 degree from our perspective, it’s basically going straight up. If you go an even smaller amount, from 0 to .00001 degrees, it’s really going straight up. The distance traveled (x) is very close to the height (sin(x)).

As x shrinks, the ratio gets closer to 100% — more motion is straight up. Radians help us see, intuitively, why sin(x)/x approaches 1 as x gets tiny. We’re just nudging along a tiny amount in a vertical direction. By the way, this also explains why sin(x) ~ x for small numbers.

Sure, you can rigorously prove this using calculus, but the radian intuition helps you understand it.

Remember, these relationships only work when measuring angles with radians. With degrees, you’re comparing your height on a circle (sin(x)) with how far some observer tilted their head (x degrees), and it gets ugly fast.

So What’s the Point?

Degrees have their place: in our own lives, we’re the focal point and want to see how things affect us. How much do I tilt my telescope, spin my snowboard, or turn my steering wheel?

With natural laws, we’re an observer describing the motion of others. Radians are about them, not us. It took me many years to realize that:

Degrees are arbitrary because they’re based on the sun (365 days ~ 360 degrees), but they are backwards because they are from the observer’s perspective.

Because radians are in terms of the mover, equations “click into place”. Converting rotational to linear speed is easy, and ideas like sin(x)/x make sense.

Even angles can be seen from more than one viewpoint, and understanding radians makes math and physics equations more intuitive. Happy math.

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166 Comments on "Intuitive Guide to Angles, Degrees and Radians"

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Tony

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Hi Khalid, I always enjoy your explanations.

While you’re talking about how arbitrary degrees are as a measure of angle, it reminded me of how radians are somewhat arbitrary. Here’s a really interesting article on how pi should actually be defined as what is currently 2 * pi:

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Hi Tony, thanks for the comment! Yep, that’s a very interesting article.

I think radians are pretty natural (distance traveled/radius), but the scale is made somewhat arbitrary (as you say) based on our definition of pi. Perhaps it would be easier if a circle had pi radians instead of 2 pi (as a result of defining pi to be the circumference of a unit circle).

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9 years 1 month ago

Fredg

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Shouldn’t the orbital speed of the satellite be the linear speed in mph divided by the distance from the center of the earth to the satellite (not the radius of the earth)?

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I’m a pilot. I can just imagine ATC telling me “Turn right heading pi radians.” Maybe they would just give me the coefficient and say turn right heading 1. Then north could be zero, east could be .5 etc. I can also imagine looking at my compass in the plane and seeing it marked with 0, 1/2pi, pi, 1 1/2 pi, 2pi, etc. Actually…no, I can’t imagine any of that at all. :)

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@Tracy: Heh, point well taken — degrees are definitely best when we’re observing our own motion :). Though a radian compass might be a fun gag.

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9 years 1 month ago

Steen

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Well from the pilot point of view he is not observing his motion but his direction in a fixed outerframe. So he is turning and then of course it will be natural to do so in degrees.

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4 months 19 days ago

Siya

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You are the only one I know who can make a math blog post sound funny!

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Thanks Siya — I think there are gems hidden away in almost any topic :).

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[…] According to Better Explained, degrees are subjective but radians are objective. A degree is the amount I, an observer, need to tilt my head to see you, the mover. It’s a tad self-centered, don’t you think? […]

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9 years 1 month ago

Sid

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Kalid I can only encourage you to write faster and write more. I can’t get enough of these explanations.

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Thanks for the encouragement Sid! I hope to increase the output too :).

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9 years 1 month ago

av

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HI KALID,
i am an comp sc grad. working for 15 yrs.
i love maths, but i had really bad maths teachers, all thru school and college.

as a result, i never understood what radians were for. THANKS A TON for this article. at last it is crystal clear.

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Thanks av, glad it was helpful! It took me quite a while to figure out what radians were about as well :).

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Kalid,
Thank you again! I agree with the others. More!! That is all long as the quality stays the same :-) and you have time to work out.

later
T.

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9 years 1 month ago

V.Manoharan

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Dear kalid,
Your site has rekindled my interest in Maths. thanks a lot.
Recently I had started assuming the following;
Degrees: Angle measured from origin
Radians: Angle measured from circumference (in terms of radius)
For equilateral triangle, the angle is 60 degrees between two sides. If these two sides are squeezed to form 57.3 degrees, third side bulges out to form an arc of a circle with 1 radian measurement.

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hi khalid!

i m a student of 11th std, and i really hated maths before meeting you, but you are a real eye opener!!

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(in continuation)…..
though i have become a great fan of yours, i just want to say that you should try to cover up topics a bit faster as you are my teacher from now onwards…..i m from india, and prep. for engineering….hope you consider it(covering up topics)…especially trigo. and quadratic equations, they are my least favourite(very tough)
thank you,
chirag

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Hi Chirag, thanks for the comment! Yep, math can be enjoyable if seen in the proper light.

I’ll try to keep cranking out posts as I can :)

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9 years 9 days ago

joe

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Your explanation of constellation rotation is incorrect. Every constellation rotates completely everyday. So if the Big Dipper is upside down it’ll be right side up in twelve hours. Now if you measure at the same time everyday each constellation will be one degree farther along than at the same time the day before and that is the once a year rotation. Your wikipedia reference has a succint description. “Ancient astronomers noticed that the stars in the sky, which circle the celestial pole every day, seem to advance in that circle by approximately one-360th of a circle, i.e., one degree, each day.”

Otherwise great post. Thanks.

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8 years 11 months ago

Jessy

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nice mathematics

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8 years 8 months ago

amit

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pl. explain something about gradient curl & divergence. these are very difficult to understand

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[…] BetterExplained has a good article on the reasons for using radians rather than degrees. This isn’t something you need to know for A-level, but it should be helpful in understanding why degrees are yanked cruelly away from you when you hit sixth form. […]

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